NONTRIVIAL SOLUTION FOR A CLASS OF SEMILINEAR BIHARMONIC EQUATION INVOLVING CRITICAL EXPONENTS

In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS OF HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS

Some theorems are obtained for the existence of nontrivial solutions of Hamiltonian systems with Lagrangian boundary conditions by the minimax methods.

NONTRIVIAL SOLUTION OF A NONLINEAR SECOND-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

In this paper, for a second-order three-point boundary value problem u″＋f（t,u）=0,0〈t〈1,au（0）-bu′（0）=0,u（1）-au（η）=0,where η∈ （0, 1）, a, b, α ∈R with a^2 ＋ b^2 〉 0, the existence of its nontrivial solution is studied. The＇conditions on f which guarantee the existence of nontrivial solution are formulated. As an application, some examples to demonstrate the results are given.

MULTIPLE NONTRIVIAL SOLUTIONS FOR A CLASS OF SEMILINEAR POLYHARMONIC EQUATIONS

In this paper, we are concerned with the following problem：{（-△）ku=λf（x）｜u｜q-2u＋g（x）｜u｜k＊-2u, x∈Ω, u∈H k0 （Ω）, where Ωis a bounded domain in RN with N ≥2k＋1, 1〈q〈2,λ〉0, f, g are continuous functions on Ω which are somewhere positive but which may change sign on Ω. k＊ = N2/N-2k is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the existence of multiple nontrivial solutions to this equation is verified.

EXISTENCE OF NONTRIVIAL SOLUTION OF QUASILINEAR ELLIPTIC EIGENVALUE PROBLEM ON R^n WITH NATURAL GROWTH CONDITIONS

In this paper, we get the existence result of the nontrivial weak solution (λ, u) of the following eigenvalue problem with natural growth conditions.

Nontrivial solutions for a class of fractional difference boundary value problems and fixed-point problems

In this work,we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:{T△^(v)_(t-1)(t△^(v)_(v-1)x(t))=f(x(t+v-1)),t∈[0,T]N_(0),x(v-2)=[tΔ^(v)_(v-1)x(t)]_(t=T=0.The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.

STRUCTURE OF NONNEGATIVE NONTRIVIAL AND POSITIVE SOLUTIONS OF SINGULARLY PERTURBED p-LAPLACE EQUATIONS

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small.

NONTRIVIAL SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR FOURTH-ORDER DIFFERENTIAL EQUATIONS

The singular boundary value problems for fourth-order differential equations are considered under some conditions concerning the first eigenvalues of the relevant linear operators. Sufficient conditions which guarantee the existence of nontrivial solutions are obtained. We use the topological degree to prove our main results.

NONTRIVIAL SOLUTIONS FOR A POLYHARMONIC EQUATION WITH SINGULAR TERM

The existence of nontrivial solutions for a polyharmonic equation with singular term is proved by Mountain Pass Lemma and Sobolev-Hardy inequality.

EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS TO p-KIRCHHOFF TYPE EQUATION

In this paper, by applying Fountain theorem, we study the existence of nontrivial solutions to the nonlinear kirchhof nonlocal equation under Ambrosetti-Rabinowitztype growth conditions.

MULTIPLE SOLUTIONS FOR NONAUTONOMOUS SECOND ORDER PERIODIC SYSTEMS

We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.

EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES

One parabolic p-Laplacian-like differential equation with mixed boundaries is au/at in the corresponding studies is replaced by a（au/at）, studied in this paper, where the item au/at which makes it more general. The sufficient condition of the existence and uniqueness of non-trivial solution in L2（O, T; L2 （Ω）） is presented by employing the techniques of splitting the boundary problems into operator equation. Compared to the corresponding work, the restrictions imposed on the equation are weaken and the proof technique is simplified. It can be regarded as the extension and complement of the previous work.

Existence of the Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term

In this paper, we deal with the existence of solution for a class of quasilinear Schrödinger equations with a nonlocal term Where μ ∈ (0,3), the function K,V ∈ C(R3,R+) and V(x) may be vanish at infinity, g is a C1 even function with g’(t) ≤ 0 for all t ≥ 0, g(0) = 1, , 0 a F is the primitive function of f which is superlinear but subcritical at infinity in the sense of Hardy-littlewood-Sobolev inequality. By the mountain pass theorem, we prove that the above equation has a nontrivial solution.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS INVOLVING m CRITICAL HARDY-SOBOLEV EXPONENTS AND m SIGN-CHANGING WEIGHT FUNCTION

In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign- changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION

In this paper,we consider a class of fractional problem with subcritical perturbation on a bounded domain as follows:(P_(k)){(−△)^(s)u = g(x)[(u − k)+]^(q−1) +u^(2^(∗)_(s)−1), x ∈Ω ,u > 0, x∈Ω ,u = 0, x ∈Ω R^(N)\Ω. We prove the existence of nontrivial solutions u_(k) of(P_(k))for each k∈(0,∞).We also investigate the concentration behavior of the solutions u_(k) as k→∞.

EXISTENCE OF SOLUTIONS TO 2m-ORDER PERIODIC BOUNDARY VALUE PROBLEM

In this paper, we are concerned with the existence of nontrivial solutions to a 2m-order nonlinear periodic boundary value problem. By the infinite dimensional Morse theory, under some conditions on nonlinear term, we obtain that there exist at least two nontrivial solutions.

Biharmonic Equation and an Improved Hardy Inequality

An improved Hardy inequality will be proven in the present work. Using the improved Hardy inequality and variational techniques, we also discuss the existence of nontrivial solution for following the weighted eigenvalue problem: